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Copyright © Mathematics Mastery 2018-19 1
This guide is for parents/carers and any adult working with the child.
The Year 5 homework book is aimed to support children’s in-class learning. There are ten pieces of
homework, each linked to the units of work in the Year 5 programme of study. The tasks provided
complement the work done in class and aim to provide opportunities for children to practise and
consolidate their understanding of key concepts relevant to the Year 5 curriculum. Each piece of
homework should take no more than 30 minutes to complete.
In order to support your child with the tasks, each piece of homework is accompanied by parental
guidance. The guidance also aims to provide an opportunity for you to understand the methods your child
is being taught, which may differ from methods you are familiar with. The methods used correspond to
the expectations of the National Curriculum 2014 and are the expected methods that children are
required to demonstrate understanding of. For additional support, there is also a glossary of key words at
the end of the book.
What is ‘Mastery’?
The ‘mastery approach’ to teaching mathematics is the underlying principle of Mathematics Mastery.
Instead of learning mathematical procedures by rote, we want your child to build a deep understanding of
concepts which will enable them to apply their learning in different situations. We do this by using three
key principles:
Conceptual understanding
Your child will use multiple concrete and
pictorial representations and make connections
between them. A key part of a ‘deep
understanding’ in maths is being able to
represent ideas in lots of different ways.
Mathematical language
When asked to explain, justify and prove their ideas,
your child is deepening their understanding of a
concept. The correct mathematical vocabulary is
taught from the outset and communication and
discussions are encouraged.
Mathematical thinking
Lots of opportunities are planned for your child
to investigate open questions that require them
to sort and compare, seek patterns and look for
rules. Good questioning, both for and from your
child, build a deeper understanding of maths.
34 864 - 25 423 = 9441
This is an equation We don’t round a number ‘up’
or ‘down’, we round it to the
nearest multiple of 10, 100,
1000 etc.
What is the same?
What is different? Is there a rule? Is it
always, sometimes
or never true?
2 Copyright © Mathematics Mastery 2018-19
‘What’s the question?’ If this is the answer, what could the question have been? This could be an equation or a word problem.
‘What’s wrong with this?’ Can you explain what is wrong with this and correct the error?
‘Draw it’ Draw a picture to explain or demonstrate what you have worked out.
‘Reason it’ Explain how you know.
‘Show me!’ Convince me that you are right.
Find a pattern’ Can you see a pattern (in the numbers)? Can you see a pattern in the answers? Continuing this pattern, what would happen if...? What came before? What comes next? Explain how you know.
‘What’s the same? What’s different?’ Can you find anything that is the same about these two numbers/shapes/calculations? Now can you find something that is different?
Have you found all possibilities?’ Is there more than one way of completing this? Is there more than one answer? Have you found them all?
Maths story’ Make up a real-life story using your equation/numbers or shapes.
‘Odd one out’ Find an odd one out and explain why it doesn’t fit. Could another one be the odd one out? Why?
Ideas for Depth
At the end of each homework piece there is a ‘Next Step for Depth’ question or activity for your child to
engage with which will provide further challenge to deepen their understanding of a concept. Children
should use the blank pages at the back of the book to record their answers. These challenges may be
open-ended, involve discussion and/or application to real life situations and you should encourage your
child to apply their learning to each task. All next step challenges support the development of at least
one of the principles of mastery. The ideas can also be used outside of maths homework tasks in day to
day discussions to allow opportunities for your child to see maths in everyday situations. The table
below explains the Next Step for Depth ideas.
Copyright © Mathematics Mastery 2018-19 3
Parental guidance
On every parental guidance page the unit title is located at the top, followed by an overview of the key
learning. In addition, you will see where the key learning fits in with what your child has previously learnt,
along with where the learning will be taken in subsequent units and years of study. It is important to
understand that the principle of mastery does not encourage acceleration and remember that depth of
understanding is key to your child becoming a confident mathematician who can think flexibly.
Additional information
Language use
For some homework tasks there is guidance on specific vocabulary or phrases that you and your child
should use. E.g.
The way that pupils speak and write about mathematics has been shown to have an impact on their
success in mathematics (Morgan, 1995; Gergen, 1995). Therefore, there is a carefully sequenced,
structured approach to introducing and reinforcing mathematical vocabulary throughout maths tasks. You
may find some terminology different to that which you are used to.
How to say decimal numbers
The digits after the decimal place are said as separate digits. For example, 0.42 is said “zero point four
two” and not “zero point forty two” as this can cause place value confusion, thinking that 0.42 is greater
than 0.5 because 42 is greater than 5.
You can find further information about the Mathematics Mastery programme online at
www.mathematicsmastery.org. If you have any questions regarding this homework book please speak
with your child’s class teacher.
Unit number and unit title
Key learning
Prior and future learning: where
this objective fits into the
sequence of learning over time.
A worked example and key points
to support your child.
The word ‘sum’ should only be used for calculations involving addition, e.g. the sum of 23 plus 24 is equal to 47.
45 - 32 = , 12 x 4 = , 240 ÷ 4 = are NOT ‘sums’ they are ‘equations’ or ‘calculations’.
4 Copyright © Mathematics Mastery 2018-19
Parental Guidance The first week of this unit revises the different ways in which fractions can be presented with a focus
on fractions of a whole using area models (e.g. shading in equal sections in shapes) and fractions as
numbers, positioning them on number lines. Pupils identify equivalent fractions and compare and
order fractions.
Prior learning In previous years, pupils began to learn about how fractions can be numbers in their own right, and how they can
also be operators (you can find a fraction of something). In Year 3 and 4, pupils began to recognise and show, using
diagrams, families of common equivalent fractions.
Future learning They will learn about percentage. More in Year 6.
Worked examples
What is a fraction?
Fractions are complex and cover a range of concepts. Ask yourself, “what is a fraction? When are fractions used?”
and you will have lots of different answers.
A fraction can describe part of a whole A fraction is a number
A fraction can be a part of a set or an amount A fraction is the result of division
Unit 6: Fractions and decimals (week 1 of 3)
How to write a fraction
The traditional way of writing fraction (starting with the numerator and working down) doesn’t look like a
fraction until the last digit. Teachers and pupils are encouraged to record a fraction in the following way:
First draw the vinculum to show that it is a fraction (the vinculum is the
horizontal line separating the numerator from the denominator)
Then write the denominator to show how many equal parts make up the whole
Finally, write the numerator to show how many of those equal parts we are
counting or are referring to. 5
3vinculum
numerator
denominator
Equivalent fractions
Fraction walls are a useful way to build understanding of
equivalent fractions. Looking at the bars in the fraction wall,
we can see, for example, that two quarters is the same as one
half and is also equal to four eights.
Through lots of experience, pupils understanding that any
fraction can be written in many different ways.
8
4
4
2
2
1
0 1 2
1
4
1
4
1= 1 ÷ 4
0 . 2 5
4 1 . 0 0 1 2
One quarter of the area of the shape is shaded
One quarter of the counters are grey
One quarter of £4 is equal to £1
I want to eat a quarter of the whole pizza
decimal equivalents
will be explored later
Copyright © Mathematics Mastery 2018-19 5
Next Step for Depth
How many different ways can you represent this fraction?
How will you explain each representation?
1) What fraction of each shape is shaded?
___________ ___________ ___________ ___________
2) Shade in the given fraction of the area of each shape.
3) Identify the fraction that each arrow points to and place these fractions on the number line.
Can you place any other fractions?
Pupil tasks
8
3
3
1
5
3
4
3
4
3
5
3
10
3
8
4
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Parental Guidance The focus of this week is the equivalence of fractions and decimals as they are different
representations of the same thing and should not be seen as separate. Children represent fractions and
decimal numbers in a variety of ways and write decimal numbers as fractions. This week introduces
children to numbers with up to three decimal places, extending to thousandths for the first time. They
develop an understanding of the relationships between ones, tenths, hundredths and thousandths.
Prior learning
Decimal notation was introduced in Year 4 and pupils worked with tenths and hundredths. Future learning Later in Year 5 pupils extend calculation strategies and methods to include decimal number and use all four
operations to solve problems involving decimals and fractions.
Worked examples
Pupils write decimal numbers as fractions, recognising and understanding the relationship between tenths,
hundredths and thousandths. They represent decimal numbers in a variety of different ways using a variety of
different equipment. A bead string with 100 beads is a useful way to represent the relationship between ones,
tenths and hundredths.
Connections can be made from the bead string to the number
line.
Unit 6: Fractions and decimals (week 2 of 3)
10
1
100
1
Dienes blocks are used to build an understanding of the relationship between ones, tenths, hundredths and
thousandths.
ones tenths hundredths thousandths
Decimal numbers can be represented with place value counters
on a place value chart or by drawing dots.
0.125 has one tenth, two hundredths and 5 thousandths.
The digits after the decimal place are said
as separate digits. For example, 0.25 is said
“zero point two five” and not “zero point
twenty five” as this can cause place value
confusion, thinking that 0.25 is greater than
0.3 because 25 is greater than 3.
1 0.1 0.01 0.001
Ten tenths is equal to one.
If the whole bead string represents one, what is the value of ten beads?
One has been divided into ten equal parts and each part has a value of one tenth.
If the whole bead string represents one, what is the value of one bead?
One has been divided into one hundred equal parts and each part has a value of one hundredth.
One hundred hundredths is equal to one.
The large cube
can represent
one whole.
This whole can
be regrouped
as ten tenths.
Each tenth can be
regrouped as ten
hundredths.
Each hundredth can be
regrouped as ten
thousandths.
Copyright © Mathematics Mastery 2018-19 7
Next Step for Depth
Choose your own decimal numbers and represent them in las many
different ways as you can.
What do the different representations show you about the number?
Pupil tasks
1) Label each diagram showing the value as a fraction and as a decimal.
a) If the whole bead string represents 1, what is the value of twenty three beads?
b) If the whole bead string represents 1, what is the value of seventy five beads?
2) Represent the number in a variety of ways. 0.125
Ones tenths hundredths thousandths
There is one tenth, __________ hundredths and _________ thousandths.
The number is said as zero point _______________________________.
10 100 + + = 1000
Dienes blocks:
Counters:
0. = _____ + 0.02 + _______
10 + = 1000
Dienes blocks: Ones tenths hundredths thousandths
Counters:
0. = _____ + _______
There are ________ tenths and _________ thousandths. The is a place holder in the ____________ place.
The number is said as zero point _______________________________.
3) Represent the number in a variety of ways. 0.405
8 Copyright © Mathematics Mastery 2018-19
Parental Guidance This week pupils order and compare decimals and develop their understanding of the relative size of
these numbers relating this to strategies used when ordering whole numbers. They then apply their
understanding of equivalent decimals and fractions to compare and order decimals and fractions. A
sense of the relative size of fractions and decimals is further developed through tasks requiring pupils
to place numbers on number lines.
Prior learning Pupils have strategies in place for ordering whole numbers and have spent time considering the relative size of
fractions. In Year 4 they ordered and compared decimals with up to two decimal places. Pupils will draw on their
understanding of equivalent fractions and decimals. as well as place value understanding.
Future learning Pupils continue to develop an understanding of the relative size of fractions and decimals. This work will relate to
understanding of percentages.
Worked examples
Comparing and ordering decimals
1) Order the numbers 0.542, 0.5 and 0.54
An understanding of place value is key to reasoning
about the relative size of decimal numbers. Representing
the numbers on a place value chart can support this.
Possible statements that could be said:
Ordering decimals and fractions on a number line
Pupils should draw on their multiple representations of fractions and decimals to support decision making when
ordering. Pupils will use a range of number lines with different values at the start and end and should consider the
value of each marking. When deciding how to order, finding a decimal equivalent of a fraction or converting all into
fractions with the same denominator are useful strategies. The focus should be on explaining their reasons.
Unit 6: Fractions and decimals (week 3 of 3)
20
13
There are more digits
in 0.542 so it must be
the largest number
542 is greater than 54
so 0.542 must be
greater than 0.54
ones tenths hundredths thousandths
All of the numbers
have five tenths 0.54 and 0.542 both
have five tenths and
four hundredths and
so both are greater
than 0.5
0.542 has two
thousandths more
than 0.54 and so is
greater
I know that twelve twentieths is equivalent to six
tenths and fourteen twentieths is equivalent to
seven tenths so this must be half way between.
0.5 1
I know that thirteen twentieths
is equivalent to 65 hundredths 0.875
I know that this is 0.125 away from one.
0.125 is equivalent to one eighth so
0.875 must be seven eighths.
Copyright © Mathematics Mastery 2018-19 9
Pupil tasks
Next Step for Depth
Clearly explain your answers to question 2 using representations of the
decimals and fractions to support you decisions.
1) Choose a symbol to make each statement correct.
is greater than is less than is equal to
6
5
5
2
0.32 0.324 0.761 0.716 0.053 0.05
> < =
0.625 0.405 0.373 1000
373
2) Choose a symbol to make each statement correct.
is greater than is less than is equal to
> < =
3) Place these fractions and decimals on the number line
3 65
20 100 0.3 0.825
0 1
Choose some interesting fractions and decimals to place on the number line.
10 Copyright © Mathematics Mastery 2018-19
Parental Guidance Pupils revise their understanding of angles including acute and obtuse angles and are introduced to
reflex angles as those greater than 180°. They use this knowledge to identify and compare angles in a
range of situations. Pupils will learn to use a protractor to measure and draw angles in degrees. Pupils
identify that angles at a point total 360° and angles at a point on a straight line total 180°.
Prior learning In Year 3 pupils were introduced to right angles and identified right angles in shapes and in every day life. This was built upon in Year 4 where pupils compared and ordered angles as well as beginning to investigate angles within shapes Future learning Pupils continue to develop their accuracy when measuring and drawing angles. In Year 6, pupils explore angles within triangles and use protractors to construct two dimensional shapes. Worked example
Pupils have been introduced to angles as a measure of the amount of the rotation. They can experience
this by standing and rotating various amounts (e.g. quarter turn clockwise, half turn anti-clockwise) and
connecting to their knowledge of direction and compass points. Imagining lines showing the direction
they are pointing before and after they have rotated allows them to
visualise the angles they are working with drawn on a page.
The angle is the amount of turn between each arm.
A common mistake made is thinking that longer lines mean a larger angle.
Children identify, compare and classify acute, obtuse or reflex angles and
definitions of these terms are in the glossary at the back.
A protractor can be used to measure degrees accurately. More often than not a protractor has two scales
(one clockwise and one anti-clockwise) going from 0 to 180 degrees.
Unit 7: Angles (week 1 of 2)
Place the centre of the protractor on the vertex
Line up one arm with the “0” on the scale (decide
which scale to follow at this stage)
Follow your chosen scale to the other arm and
read the angle
The most common mistakes are misaligning
the centre of the protractor and following the
wrong scale. Encourage children to estimate
the angle prior to measuring it, reducing the
risk of using the wrong scale.
Copyright © Mathematics Mastery 2018-19 11
Pupil tasks
Next Step for Depth
Make a poster warning against all the possible mistakes that can easily be made
when using a protractor. Include Top Tips for accurate measuring with a protractor.
1) Complete the descriptions of each type of angle.
An acute angle is A right angle An obtuse angle is between A reflex angle is
less than ______ is equal to _____ ______ and ______ greater than _____
2) For each angle, match them to the correct size.
10° 100° 340° 45° 160°
3) Estimate each angle and then use a protractor to measure and check your estimate.
Estimate of angle________________
Measurement of angle _____________
Estimate of angle________________
Measurement of angle _____________
12 Copyright © Mathematics Mastery 2018-19
Parental Guidance Pupils continue to identify and compare acute, obtuse and reflex angles as well as angles that are
multiples of 90°. Through continued practice pupils increase accuracy when measuring angles and
explore drawing angles using a ruler and a protractor. Pupils use their knowledge that a whole turn is
360° and half a turn is 180° to explore angles that meet at a point and on a straight line.
Encourage pupils to explore angles in real life for example looking at patterns or architecture.
Prior learning See previous week
Future learning
See previous week
Worked example
Some suggested questions you could explore to review the learning across this unit:
“What is an angle?” “What unit of measure do we use for angles?”
“When is one angle ‘bigger’ than another?” “What is the same and what’s different?”
Unit 7: Angles (week 2 of 2)
This angle is 70°
I know a full turn is 360° so I
can work out the reflex angle
by calculating 70 + = 360
360 - 70 = 290 So this angle must be 290°
Measuring reflex angles
How can you use a semi-circular protractor to
measure a reflex angle? It isn’t big enough! The
angle is larger than the scale on the protractor.
The reflex angle can be worked out by measuring
the other angle and subtracting that from 360.
Angles at a point add up to 360°
To work out the size of angle a, without measuring, you
can find the value that makes all the angles total a value
of 360. This can be represented with this bar model.
Angles on a straight line add up to 180°
To work out the size of angle b, without measuring,
you can find the value that makes both angles sum to
180. A bar model can be used to represent this.
When measuring angles drawn like this, think
carefully about what is being measured as there is
more than one angle as well as more than one scale
on the protractor and it is easy to make an error.
Copyright © Mathematics Mastery 2018-19 13
Pupil tasks 1) Work out the size of each of these reflex angles.
2) Draw and label the following (use the notes page at the back if there is not enough room).
a) an angle of 27° b) an angle of 132° c) an angle of 250°
3) Work out the missing angles. Do not measure, instead use your knowledge of angles.
Next Step for Depth
What are possible values of the three angles? Draw them at the back of the book.
How many different answers can you find? Can you find them all?
There are three angles on a straight line.
One of them is a right angle.
Another is a multiple of 5.
a° = ° b° = ° c° = °
a° = °
a°
b°
b° = °
14 Copyright © Mathematics Mastery 2018-19
Parental Guidance Pupils calculate with fractions and decimals. They start with addition and subtraction of fractions with
the same denominator, and then fractions with denominators that are multiples of the same number.
They multiply fractions by a whole number and connect this to finding fractions of an amount.
Prior learning In Year 3 and 4, pupils added fractions with the same denominator. They developed an understanding of equivalent fractions
Future learning
Pupils continue to calculate with fractions and decimals in a range of contexts throughout Years 5 and 6.
Worked example
Adding and subtracting fractions
Pupils add and subtract fractions starting with fractions with the same denominator. Pupils then add and subtract
fractions where the denominator of one fraction is a multiple of the other (e.g. half add eighths) by identifying
equivalent fractions with the same denominator. They become fluent through a variety of increasingly complex
problems, supported with images and concrete equipment.
Improper fractions and mixed numbers
Fractions can be greater than one.
An improper fraction has a numerator that is greater than the denominator.
A mixed number is a whole number and a fraction.
Pupils will add and subtract fractions, including improper fractions and mixed
numbers, be expected to record an improper fraction as a mixed number and
the other way around.
Multiplying a fraction by a whole number
Unit 8: Fractions, decimals and percentages (week 1 of 3)
4
3
4
1
2
1
8
3
4
1
8
5
4
1
8
5
?
2
1
4
1
4
1
2
1
= 1 8
11
8
3
I used of
the ribbon. 5
1
I used three
times as much.
5
1 x 3 = 5
3
Copyright © Mathematics Mastery 2018-19 15
Pupil tasks
1) Use the bar models to show the answer to each calculation.
2) Doris eats five eighths of one pizza and half of another pizza.
This diagram shows what was left after she’d eaten: How much pizza did she eat altogether?
Show on this diagram:
Record as an improper fraction: Record as a mixed number:
3) A mug holds of a litre. How much will seven mugs hold?
Record as an improper fraction and a mixed number.
Next Step for Depth four fifths less than one and a half
8
3
2
1
2
1
8
5
=
=
5
4
2
11
fourteen out
of 20 objects one half plus one fifth
× 7 10
1
Doris ate of the pizza. Doris ate of the pizza.
4
1
16 Copyright © Mathematics Mastery 2018-19
Parental Guidance This week pupils think about fractions as an operator (you can find a fraction of something) as well as
thinking of a fraction as the result of division. They connect this to their understanding of
multiplication and explore a range of contexts in which they multiply a fraction by a whole number.
Prior learning Pupils have been doubling and halving numbers since Year 1 and have connected this to an understanding of
fractions as operators. This is extended to finding other unit fractions and then non-unit fractions of amounts.
Future learning Pupils continue to calculate with fractions and decimals and develop an understanding of percentages in a range of
contexts throughout Years 5 and 6.
Worked examples
Fractions of an amount
Fractions can be seen as an operator and you can find a fraction of an amount. This is introduced through doubling
and halving; developing an understanding that finding half of an amount is the same as dividing the amount by
two. Through plenty of experience with concrete materials and pictorial representations, pupils find fractions of an
amount, connecting this to their understanding of multiplication.
There are 20 people on a bus and one fifth of them are children. How many children are there on the bus?
Represent as part of the set of people: Represent with a bar model:
of 20 = 4 20 ÷ 5 = 4 × 20 = 4
Having found one fifth, pupils can extend to finding non unit
fraction e.g. two fifths or three fifths. To find a non-unit fraction
of an amount, find the unit fraction with the same denominator
and multiply this amount by the numerator.
Multiplying a fraction by a whole number
There are temptations to teach “tricks” (such as ‘just multiply the top number’) that are not based in conceptual
understanding. Pupils should have plenty of opportunities to experience calculating with fractions in a range of
contexts, to represent in a variety of ways and to explain and think carefully about what they are doing.
A small bottle holds of a litre. How much liquid will six bottles hold?
Pupils could, initially, skip count in eighths to calculate the answer.
A bottle holds of a litre. How much liquid will five bottles hold?
Connecting this to understanding of calculations with integers is useful.
Pupils can identify that 3 ones × 5 = 15 ones and so 3 eighths × 5 = 15 eighths.
This can then be converted from an improper fraction to a mixed number.
Unit 8: Fractions, decimals and percentages (week 2 of 3)
20
5
15
1
5
1
5
1
20
5
2I know that one fifth of 20 is equal
to 4 so two fifths is equal to 4 × 2 5
2of 20 = 8 5
2× 20 = 8
8
1
× 6 = 8
6
8
1
8
3
8
15× 5 =
8
3
8
1
?
?
8
3
Copyright © Mathematics Mastery 2018-19 17
Pupil tasks
1) There are 64 apples. Answer each question with either a fraction or a whole number:
a) of the apples are green and the others are red.
How many apples are green? What fraction of apples are red?
How many apples are red?
c) of the apples are rotten and the others are not.
How many apples are rotten? What fraction of apples are not rotten?
How many apples are not rotten?
c) 16 apples are in a basket and the others are in a box.
What fraction of the apples are in the basket?
What fraction of the apples are in the box?
2) Represent each problem and then calculate the answer (use the notes page at the back is needed).
a) Amy works out that her train journey takes of a day. She has a plane journey that is five times
longer. What fraction of a day will the plane journey take?
b) A jug holds of a litre of liquid. How much liquid can four jugs hold?
3) Calculate the area and the perimeter of this rectangle:
Area = cm2
Perimeter = cm
Next Step for Depth
Generate word problems involving the multiplication of a fraction by a whole
number.
4
3
8
1
12
1
3
2
18 Copyright © Mathematics Mastery 2018-19
Parental Guidance Pupils are introduced to percentage as ‘the number of parts per hundred’ and understand that
fractions, decimal fractions and percentages are different representations of the same thing. They
identify and calculate percentages in a range of situations.
Prior learning This is the first year that pupils learn about percentage.
Future learning
Pupils solve problems which require knowing equivalent fractions, decimals and percentages as well as using percentage for comparison.
Worked example
Unit 8: Fractions, decimals and percentages (week 3 of 3)
100
12
12 equal parts out of 100 equal parts are shaded, so 12% of the grid is shaded.
12% = = 0.12
5
3
100
60
10
6
5
3
5
3= 0.6
Six tenths is equal to 60 hundredths
60 equal parts out of 100 are shaded and so
60% of the
grid is shaded.
of the grid
is shaded.
When there are not 100 equal parts, use knowledge of equivalent fractions. To work out the percentage of the
grid that is shaded, work out the number of parts per hundred this is equivalent to.
What percentage of the grid is shaded?
‘Per cent’ means ‘for each hundred’ and children build the understanding that
the percentage relates to the number of parts per hundred.
When there are 100 equal parts, identifying percentage is straight forward.
What percentage of the grid is shaded?
Use equivalent fractions:
Divide the whole into 100 equal parts:
Use decimals:
Finding a percentage of an amount
Having built an understanding of fraction, decimal and percentage equivalence, pupils use understanding of
finding fractions of amounts to find percentages of amounts.
There are 200 athletes in a team. 25% of them compete in swimming events. How many swimmers are
there on the team?
I know that 25% is the
same as one quarter.
To find one quarter of 200, I
divide 200 into four equal parts.
Each part has a value of 50 so 25% of 200 = 50.
There were 50 swimmers on the team.
200
50
200
50
A bead string with 100 beads can be used to represent 100%. In this case, 100% is equal to 200 athletes so 100
beads has a value of 200 and we need to find the value of 25 beads.
100 beads have the value of 200 so 50 beads have a
value of 100 and 25 beads have a value of 50.
Copyright © Mathematics Mastery 2018-19 19
Pupil tasks
1) What percentage of each grid is shaded? Write each percentage as a fraction and a decimal.
2) Match each card to the percentage that is equivalent to the decimal or fraction. There is more than
one for each percentage.
a) 2%
b) 22%
c) 20%
3) Find percentages of amounts. Mark the bead string and complete the statements.
a) What is 20% of 50?
b) What is 75% of 80?
Next Step for Depth Which of these is the odd one out? How many different answers can you
think of?
Percentage
shaded:
Fraction
shaded:
This can represent the decimal:
Percentage
shaded:
Fraction
shaded:
This can represent the decimal:
0.2 0.22 0.02 100
20 10
2 100
2 100
22 5
1
50
80
20% of 50 =
75% of 80 =
5
3 0.5 100
6 65%
20 Copyright © Mathematics Mastery 2018-19
Parental Guidance
The first week of this unit revises how to describe positions on a 2-D grid as coordinates with both
positive and negative numbers. Pupils will identify, describe and represent the position of a shape
following a translation and know that the shape has not changed.
Prior learning In Year 4 pupils were introduced to the coordinate system and described positions that have coordinates with
positive numbers. They used a variety of language to describe position and movement.
Future learning Pupils will continue to use coordinates with positive and negative numbers to describe position and will explore
properties of shapes.
Worked examples
Coordinates The coordinate system is a way of describing position or location in two dimensions. Two number lines lie at right
angles to each other, the horizontal line is the x-axis, the vertical line is the y-axis and the point where they meet
is the origin. Any point in the space can be described with two numbers, called its coordinates.
Unit 9: Transformations (week 1 of 2)
The x-coordinate is the distance moved horizontally
along the x-axis and the y-coordinate is the distance
moved vertically along the y-axis. For example, to
reach point A we move three units along the x-axis and
then two units along the y-axis. The coordinates of
point A are (3, 2). The convention is always to give the
x-coordinate first and the y-coordinate second (x, y).
What do you notice about point D? Although decimal
coordinates are not a part of the primary curriculum, it
is worth highlighting that decimal numbers can be used
to describe any point on the grid, not just those on the
lines.
0
A (3, 2)
B (−2, 0)
C (−4, −1)
D (2.4, 0.6) x-axis
y-axis
x
y
A
B
Notice that the lines are labelled. On a map
it is often the spaces that are labelled.
Translating shapes
A translation moves a shape so that it is in a different position.
When a shape is translated it retains the same size, area,
angles and line lengths and therefore the two shapes can be
described as congruent.
Shape A has been
translated four
Following the
path of one vertex
A common error made is counting the
points and counting the vertex of A as one
leading to an incorrect answer of five.
Triangle C has been
translated three right
and two down
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1) Plot the given coordinates and join the points in order, with straight lines, to create a shape.
Describe the shape: Describe the shape:
2a) Translate this shape two left and two up 2b) Translate this shape three right and one down
Pupil tasks (page 1 of 2)
6
5
4
3
2
1
1 2 3 4 5 6 0
6
5
4
3
2
1
1 2 3 4 5 6 0 -6 -5 -4 -3 -2 -1
6
5
4
3
2
1
1 2 3 4 5 6 0
(3, 6) (2, 4) (0, 3) (2, 2)
(3, 0) (4, 2) (6, 3) (4, 4) (0, 2) (2, 0) (4, 2) (2, 4)
(4, 4) (4, 6) (−2, 2)
(−4, 3) (−5, 2) (−2, 0)
Describe the shape:
Remember to join the points in order
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5
4
3
2
1
1 2 3 4 5 6 0 -6 -5 -4 -3 -2 -1
-1
-2
-3
-4
-5
3) Which of these shapes is a translation of shape A?
Pupil tasks (page 2 of 2)
4) Plot these coordinates and join each point with a straight line segment to make a shape.
(1, 3) (−5, 3) (−5, −1) ( 1, −1)
Then translate the shape three right and two down. Record the coordinates of each vertex:
( __ , __ ) ( __ , __ ) ( __ , __ ) ( __ , __ )
Write the letter of the shape
and describe the translation.
Next Step for Depth
What pattern do you notice with the coordinates?
What is the difference between the numbers in the coordinates of the vertices of
the shape before and after it has been translated?
A B
C
D E
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Parental Guidance
The week, pupils continue to use coordinates in all four quadrants to describe position. They reflect
shapes and identify, describe and represent the position of the shape and know that the shape has not
changed: it is a congruent shape.
Prior learning
Recognising and identifying reflections is connected with previous work on symmetry and identifying lines of
symmetry as well as creating symmetrical shapes and images.
Future learning
Pupils reflect shapes in mirror lines that are diagonal including shapes that touch and cross the mirror line.
Worked examples
Reflection
Reflections are everywhere and pupils will have experienced reflections in mirrors or
water. Pupils have thought about mirror lines when they explored symmetrical shapes
and patterns.
In this picture you can see the
shape of the top of the mountain
reflected in the water.
Every point is the same distance
from the mirror line.
The mirror line can be horizontal or vertical. In fact, it can be in any direction as shown in the examples below.
Pupils may have difficulties with shapes that touch the mirror line.
Pattern seeking with coordinates
When shapes are plotted on a coordinate grid and are
reflected in the x-axis or y-axis, this reinforces the fact
that each point is the same distance from the mirror
line. Pupils can seek patterns in the coordinates for each
vertex of the shape.
Unit 9: Transformations (week 2 of 2)
(-3, 3) (-1, 3)
(-1, 1) (-3, 1)
(1, 3) (3, 3)
(3, 1) (1, 1) ? What do you notice about the coordinates of this
rectangle that has been reflected in the y-axis?
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1) Reflect each shape in the dotted mirror line.
2) Reflect shape A in the y-axis and label it shape B. Then reflect shape A and shape B in the x-axis
and label them C and D. Record the coordinates of the vertices of each shape.
6
5
4
3
2
1
1 2 3 4 5 6 0 -6 -5 -4 -3 -2 -1
-1
-2
-3
-4
-5
-6
Pupil tasks
Coordinates of
shape A:
Coordinates of
shape B:
Coordinates of
shape D:
Coordinates of
shape C:
A
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Next Step for Depth
For the pattern you have created in question 3, is there
a different way to create the same design?
Pupil tasks
6
5
4
3
2
1 2 3 4 5 6 0 -6 -5 -4 -3 -2 -1
-1
-2
-3
-4
-5
-6
3) On the grid below, create a pattern by drawing a shape and translating and reflecting. In the space
below, describe what you did.
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Notes pages
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Notes pages
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Notes pages These hundred grids are useful when exploring the equivalence between fractions, decimals and percentages.
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Notes pages
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Notes pages
6
5
4
3
2
1
1 2 3 4 5 6 0 -6 -5 -4 -3 -2 -1
-1
-2
-3
-4
-5
-6
6
5
4
3
2
1
1 2 3 4 5 6 0 -6 -5 -4 -3 -2 -1
-1
-2
-3
-4
-5
-6
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Notes pages
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Glossary
Word Definition Example
proper
fraction
A fraction that has a numerator smaller than the
denominator
unit
fraction A fraction where the numerator is one
improper
fraction
A fraction that has a numerator greater than the
denominator.
mixed
number
A number written as a whole number with a
fraction
5
1
numerator
denominator vinculum
= 1 5
6
5
1
Word Definition Example
polygon 2-D shapes with three or more straight sides.
regular
polygon
A regular polygon has all sides of equal length and
all angles equal.
a square is a regular rectangle
and a regular triangle is an
equilateral triangle.
angle
The amount of turn between two straight lines that
end at the same point (a vertex). Measured in
degrees.
See below
Acute
Less than 90°
Right
90°
Obtuse
Greater than 90°
Less than 180°
Straight
180°
Reflex
Greater than 180°
5
2
4
1
3
1
5
6
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